A measurement of the $CP$ asymmetry in ${D}^{0}\ensuremath{\rightarrow}{K}_{\mathrm{S}}^{0}{K}_{\mathrm{S}}^{0}$ decays is reported, based on a data sample of proton-proton collisions collected by the LHCb experiment from 2015 to 2018, corresponding to an integrated luminosity of $6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The flavor of the ${D}^{0}$ candidate is determined using the charge of the ${D}^{*\ifmmode\pm\else\textpm\fi{}}$ meson, from which the decay is required to originate. The ${D}^{0}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ decay is used as a calibration channel. The time-integrated $CP$ asymmetry for the ${D}^{0}\ensuremath{\rightarrow}{K}_{\mathrm{S}}^{0}{K}_{\mathrm{S}}^{0}$ mode is measured to be ${\mathcal{A}}^{CP}({D}^{0}\ensuremath{\rightarrow}{K}_{\mathrm{S}}^{0}{K}_{\mathrm{S}}^{0})=(\ensuremath{-}3.1\ifmmode\pm\else\textpm\fi{}1.2\ifmmode\pm\else\textpm\fi{}0.4\ifmmode\pm\else\textpm\fi{}0.2)$, where the first uncertainty is statistical, the second is systematic, and the third is due to the uncertainty on the $CP$ asymmetry of the calibration channel. This is the most precise determination of this quantity to date.